8 1. Differential Equations curately describe any part of the real universe. Again, it might be best to think of these differential equations as being like the laws of physics, determining what is possible, and their solutions as being those things which can possibly happen under those laws, but in some hypothetical universe governed by the equation being considered. By studying equations in the abstract, we can make discoveries about what differential equations can and cannot do, and that in turn can guide our understanding of the real universe. (In fact, as Chapter 3 will explain, soliton theory itself grew out of a realization that differ- ential equations could do things which had previously been considered impossible.) 1.4 Named Equations Those differential equations which are of special interest often are known by a name, usually “the Foo equation”, where “Foo” is ei- ther the names of researchers who have worked on it or some sort of description of the application of the equation. This name is then recognized and appreciated by experts in the fields for which it is relevant. For example: The equation from thermodynamics which describes how varia- tions in temperature will diffuse through an object as time passes (mentioned in the previous section) is called “The Heat Equation”. “Maxwell’s Equations” describe electromagnetic waves (such as light or radio transmissions). “The Lotka-Volterra Equations” (also known as “predator-prey equations”) model how the sizes of two ideal populations of animals will change in time. In the next chapter we will encounter two important and famous named equations: D’Alembert’s Wave Equation (2.2) and the Inviscid Burger’s Equation (2.7). Both of these are used in the physical sci- ences to model wave phenomena, and it will be equations that model waves (whether water waves, sound waves, light waves, or even the mysterious wave-like nature of particles in quantum physics) that will be the primary focus of this book. In fact, one equation in particular will be the sole differential equation of interest for several chapters: the KdV Equation (3.1) which was originally written to model surface
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